a) If $r$$r$ is the radius of a circle then: $\displaystyle \text{ Perimeter } = 2 \pi r$$\displaystyle \text{ Perimeter } = 2 \pi r$ $\displaystyle \text{Area } = \pi r^2 \text{ or Area } = \frac{1}{4} \pi d^2$$\displaystyle \text{Area } = \pi r^2 \text{ or Area } = \frac{1}{4} \pi d^2$ $\displaystyle \text{Diameter } (d) = 2r$$\displaystyle \text{Diameter } (d) = 2r$ b) If $r$$r$ is the radius of a circle then: $\displaystyle \text{Perimeter of a semi circle } = \pi r + 2r = (\pi + 2)r$$\displaystyle \text{Perimeter of a semi circle } = \pi r + 2r = (\pi + 2)r$ $\displaystyle \text{Area of semi circle } = \frac{1}{2} \pi r^2$$\displaystyle \text{Area of semi circle } = \frac{1}{2} \pi r^2$ c) If $r$$r$ is the radius of a circle then: $\displaystyle \text{Perimeter of the quadrant } = \frac{1}{4} \times 2 \pi r + 2 r = ( \frac{\pi}{2} + 2)r$$\displaystyle \text{Perimeter of the quadrant } = \frac{1}{4} \times 2 \pi r + 2 r = ( \frac{\pi}{2} + 2)r$ $\displaystyle \text{Area of the quadrant } = \frac{1}{4} \pi r^2$$\displaystyle \text{Area of the quadrant } = \frac{1}{4} \pi r^2$ d) If $R$$R$ and $r$$r$ are radii of two concentric circles, then $\displaystyle \text{ Area enclosed between the two circle } \\ \\ = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) = \pi (R+r)(R-r)$$\displaystyle \text{ Area enclosed between the two circle } \\ \\ = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) = \pi (R+r)(R-r)$

e) If  $\triangle ABC$ is an equilateral triangle of side $\displaystyle a \text{ and height } h \text{ , then } h = \frac{\sqrt{3}}{2} a$.

Also if $R$ and $r$ are the radii of the circumscribed and inscribed circles of $\displaystyle \triangle ABC$, then $\displaystyle R = \frac{2h}{3} \text{ and } r= \frac{h}{3}$

$\displaystyle \text{Circumference of the circumcircle of } \triangle ABC = 2\pi R = 2 \pi \times \frac{2h}{3} + \frac{4}{3} \pi h$

$\displaystyle \text{Area of the circumcircle } = \pi R^2 = \pi ( \frac{2h}{3} )^2 = \frac{4}{9} \pi h^2$

$\displaystyle \text{Circumference of the inscribed circle } = 2 \pi r = \frac{2 \pi}{3} h$

$\displaystyle \text{Area of the inscribed circle } = \pi r^2 = \frac{1}{9} \pi h^2$

f) Important notes:

• If two circles touch internally, then the distance between their centers is equal to the difference of their radii.
• If two circles touch externally. Then the distance between their centers is equal to the sum of their radii.
• The distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.
• The number of revolutions completed by a rotating wheel in one minute $\displaystyle = \frac{\text{Distance moved in one minute}}{\text{circumference}}$